LMFDB - Embedded newform 804.2.ba.a.281.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(41,804)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([0, 33, 53]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("804.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.ba (of order $66$, degree $20$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{66}]$

Embedding invariants

Embedding label			281.1
Root			$-0.327068 - 0.945001i$ of defining polynomial
Character	$\chi$	$=$	804.281
Dual form			804.2.ba.a.701.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.30900 + 1.13425i) q^{3} +(1.65416 - 4.13190i) q^{7} +(0.426945 + 2.96946i) q^{9} +O(q^{10})$	$q+(1.30900 + 1.13425i) q^{3} +(1.65416 - 4.13190i) q^{7} +(0.426945 + 2.96946i) q^{9} +(-0.540736 - 5.66284i) q^{13} +(7.29407 - 2.92011i) q^{19} +(6.85191 - 3.53240i) q^{21} +(-2.07708 + 4.54816i) q^{25} +(-2.80925 + 4.37128i) q^{27} +(0.426797 - 4.46962i) q^{31} +(1.07398 + 1.86019i) q^{37} +(5.71527 - 8.02597i) q^{39} +(-3.53426 + 12.0366i) q^{43} +(-9.27019 - 8.83911i) q^{49} +(12.8601 + 4.45091i) q^{57} +(10.9516 + 0.521689i) q^{61} +(12.9758 + 3.14788i) q^{63} +(3.22957 + 7.52130i) q^{67} +(-0.120810 + 2.53611i) q^{73} +(-7.87764 + 3.59760i) q^{75} +(-13.7353 + 9.78084i) q^{79} +(-8.63544 + 2.53559i) q^{81} +(-24.2927 - 7.13299i) q^{91} +(5.62836 - 5.36663i) q^{93} +(14.0777 - 8.12776i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^7 + 6 * q^9	$ 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) $ 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$e\left(\frac{19}{66}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.30900	+	1.13425i	0.755750	+	0.654861i
$3$	1.30900	+	1.13425i	0.755750	+	0.654861i
$4$		0			0
$5$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$5$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$6$		0			0
$7$	1.65416	−	4.13190i	0.625215	−	1.56171i	−0.189520	−	0.981877i	$-0.560693\pi$
$7$	1.65416	−	4.13190i	0.625215	−	1.56171i	0.814735	−	0.579834i	$-0.196882\pi$
$8$		0			0
$9$	0.426945	+	2.96946i	0.142315	+	0.989821i
$10$		0			0
$11$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$11$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$12$		0			0
$13$	−0.540736	−	5.66284i	−0.149973	−	1.57059i	−0.686768	−	0.726876i	$-0.740971\pi$
$13$	−0.540736	−	5.66284i	−0.149973	−	1.57059i	0.536795	−	0.843712i	$-0.319635\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$17$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$18$		0			0
$19$	7.29407	−	2.92011i	1.67338	−	0.669918i	0.675643	−	0.737229i	$-0.263866\pi$
$19$	7.29407	−	2.92011i	1.67338	−	0.669918i	0.997732	+	0.0673101i	$0.0214417\pi$
$20$		0			0
$21$	6.85191	−	3.53240i	1.49521	−	0.770834i
$22$		0			0
$23$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$23$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$24$		0			0
$25$	−2.07708	+	4.54816i	−0.415415	+	0.909632i
$26$		0			0
$27$	−2.80925	+	4.37128i	−0.540641	+	0.841254i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	0.426797	−	4.46962i	0.0766551	−	0.802768i	−0.872636	−	0.488371i	$-0.837591\pi$
$31$	0.426797	−	4.46962i	0.0766551	−	0.802768i	0.949291	−	0.314398i	$-0.101803\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.07398	+	1.86019i	0.176562	+	0.305814i	0.940701	−	0.339238i	$-0.110169\pi$
$37$	1.07398	+	1.86019i	0.176562	+	0.305814i	−0.764139	+	0.645052i	$0.776836\pi$
$38$		0			0
$39$	5.71527	−	8.02597i	0.915175	−	1.28518i
$40$		0			0
$41$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$41$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$42$		0			0
$43$	−3.53426	+	12.0366i	−0.538970	+	1.83556i	0.0103449	+	0.999946i	$0.496707\pi$
$43$	−3.53426	+	12.0366i	−0.538970	+	1.83556i	−0.549315	+	0.835616i	$0.685111\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$47$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$48$		0			0
$49$	−9.27019	−	8.83911i	−1.32431	−	1.26273i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$53$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	12.8601	+	4.45091i	1.70336	+	0.589537i
$58$		0			0
$59$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$59$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$60$		0			0
$61$	10.9516	+	0.521689i	1.40221	+	0.0667954i	0.734996	−	0.678072i	$-0.237184\pi$
$61$	10.9516	+	0.521689i	1.40221	+	0.0667954i	0.667213	+	0.744867i	$0.267487\pi$
$62$		0			0
$63$	12.9758	+	3.14788i	1.63479	+	0.396596i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.22957	+	7.52130i	0.394555	+	0.918873i
$67$	3.22957	+	7.52130i	0.394555	+	0.918873i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$71$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$72$		0			0
$73$	−0.120810	+	2.53611i	−0.0141397	+	0.296829i	0.980710	+	0.195468i	$0.0626226\pi$
$73$	−0.120810	+	2.53611i	−0.0141397	+	0.296829i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	−7.87764	+	3.59760i	−0.909632	+	0.415415i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−13.7353	+	9.78084i	−1.54534	+	1.10043i	−0.589013	+	0.808124i	$0.700483\pi$
$79$	−13.7353	+	9.78084i	−1.54534	+	1.10043i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	−8.63544	+	2.53559i	−0.959493	+	0.281733i
$82$		0			0
$83$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$83$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$89$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$90$		0			0
$91$	−24.2927	−	7.13299i	−2.54657	−	0.747741i
$92$		0			0
$93$	5.62836	−	5.36663i	0.583633	−	0.556493i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	14.0777	−	8.12776i	1.42937	−	0.825249i	0.432302	−	0.901729i	$-0.357701\pi$
$97$	14.0777	−	8.12776i	1.42937	−	0.825249i	0.997071	+	0.0764798i	$0.0243681\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$101$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$102$		0			0
$103$	0.579907	+	0.0553745i	0.0571400	+	0.00545621i	0.123587	−	0.992334i	$-0.460560\pi$
$103$	0.579907	+	0.0553745i	0.0571400	+	0.00545621i	−0.0664469	+	0.997790i	$0.521166\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$107$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$108$		0			0
$109$	−15.9710	−	7.29371i	−1.52974	−	0.698611i	−0.540027	−	0.841648i	$-0.681586\pi$
$109$	−15.9710	−	7.29371i	−1.52974	−	0.698611i	−0.989717	+	0.143037i	$0.954313\pi$
$110$		0			0
$111$	−0.704087	+	3.65315i	−0.0668290	+	0.346742i
$112$		0			0
$113$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$113$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	16.5847	−	4.02341i	1.53326	−	0.371965i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.9502	−	1.04562i	0.995472	−	0.0950560i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.91119	−	1.16547i	−0.258326	−	0.103418i	0.238885	−	0.971048i	$-0.423218\pi$
$127$	−2.91119	−	1.16547i	−0.258326	−	0.103418i	−0.497211	+	0.867630i	$0.665643\pi$
$128$		0			0
$129$	−18.2789	+	11.7471i	−1.60936	+	1.03428i
$130$		0			0
$131$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$131$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$132$		0			0
$133$		−	34.9687i		−	3.03217i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$137$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$138$		0			0
$139$	−7.55788	−	11.7603i	−0.641051	−	0.997495i	−0.997990	−	0.0633672i	$-0.979816\pi$
$139$	−7.55788	−	11.7603i	−0.641051	−	0.997495i	0.356939	−	0.934128i	$-0.383820\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.10887	−	22.0851i	−0.173937	−	1.82155i
$148$		0			0
$149$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$149$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$150$		0			0
$151$	5.66550	+	23.3535i	0.461052	+	1.90048i	0.434141	+	0.900845i	$0.357052\pi$
$151$	5.66550	+	23.3535i	0.461052	+	1.90048i	0.0269113	+	0.999638i	$0.491433\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−22.1577	−	4.27054i	−1.76837	−	0.340826i	−0.801875	−	0.597492i	$-0.796164\pi$
$157$	−22.1577	−	4.27054i	−1.76837	−	0.340826i	−0.966500	+	0.256666i	$0.917376\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−12.7256	+	22.0414i	−0.996748	+	1.72642i	−0.428587	+	0.903501i	$0.640988\pi$
$163$	−12.7256	+	22.0414i	−0.996748	+	1.72642i	−0.568161	+	0.822917i	$0.692345\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$167$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$168$		0			0
$169$	−19.0103	+	3.66393i	−1.46233	+	0.281841i
$170$		0			0
$171$	11.7853	+	20.4128i	0.901246	+	1.56100i
$172$		0			0
$173$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$173$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$174$		0			0
$175$	15.3567	+	16.1057i	1.16086	+	1.21747i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$179$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$180$		0			0
$181$	−8.77893	−	25.3651i	−0.652533	−	1.88537i	−0.392380	−	0.919803i	$-0.628348\pi$
$181$	−8.77893	−	25.3651i	−0.652533	−	1.88537i	−0.260153	−	0.965567i	$-0.583773\pi$
$182$		0			0
$183$	13.7439	+	13.1048i	1.01598	+	0.968732i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.4147	+	18.8384i	0.975778	+	1.37029i
$190$		0			0
$191$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$191$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$192$		0			0
$193$	6.46337	+	14.1528i	0.465243	+	1.01874i	0.986261	+	0.165195i	$0.0528253\pi$
$193$	6.46337	+	14.1528i	0.465243	+	1.01874i	−0.521018	+	0.853546i	$0.674447\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$197$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$198$		0			0
$199$	−17.8081	+	14.0044i	−1.26238	+	0.992748i	−0.262770	+	0.964858i	$0.584636\pi$
$199$	−17.8081	+	14.0044i	−1.26238	+	0.992748i	−0.999611	+	0.0278890i	$0.991121\pi$
$200$		0			0
$201$	−4.30355	+	13.5085i	−0.303549	+	0.952816i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	8.28554	−	23.9395i	0.570400	−	1.64806i	−0.177400	−	0.984139i	$-0.556769\pi$
$211$	8.28554	−	23.9395i	0.570400	−	1.64806i	0.747800	−	0.663924i	$-0.231110\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−17.7620	−	9.15697i	−1.20577	−	0.621615i
$218$		0			0
$219$	−3.03472	+	3.18273i	−0.205068	+	0.215069i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.7168	+	19.2922i	1.11944	+	1.29190i	0.952027	+	0.306015i	$0.0989957\pi$
$223$	16.7168	+	19.2922i	1.11944	+	1.29190i	0.167412	+	0.985887i	$0.446459\pi$
$224$		0			0
$225$	−14.3924	−	4.22599i	−0.959493	−	0.281733i
$226$		0			0
$227$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$227$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$228$		0			0
$229$	−18.5574	−	13.2147i	−1.22631	−	0.873252i	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−18.5574	−	13.2147i	−1.22631	−	0.873252i	−0.995024	+	0.0996338i	$0.968233\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$233$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−29.0733	−	2.77617i	−1.88852	−	0.180331i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−13.9403	−	8.95889i	−0.897974	−	0.577093i	0.00821459	−	0.999966i	$-0.497385\pi$
$241$	−13.9403	−	8.95889i	−0.897974	−	0.577093i	−0.906189	+	0.422873i	$0.861022\pi$
$242$		0			0
$243$	−14.1798	−	6.47568i	−0.909632	−	0.415415i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−20.4803	−	39.7262i	−1.30313	−	2.52772i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$251$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$257$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$258$		0			0
$259$	9.46267	−	1.36053i	0.587981	−	0.0845390i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$263$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	23.3703	+	20.2504i	1.41964	+	1.23013i	0.934631	+	0.355619i	$0.115730\pi$
$271$	23.3703	+	20.2504i	1.41964	+	1.23013i	0.485012	+	0.874508i	$0.338815\pi$
$272$		0			0
$273$	−23.7085	−	36.8911i	−1.43490	−	2.23275i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3.74693	+	26.0605i	0.225131	+	1.56582i	0.718203	+	0.695833i	$0.244965\pi$
$277$	3.74693	+	26.0605i	0.225131	+	1.56582i	−0.493072	+	0.869989i	$0.664126\pi$
$278$		0			0
$279$	13.4546	−	0.640922i	0.805506	−	0.0383710i
$280$		0			0
$281$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$281$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$282$		0			0
$283$	−4.63073	+	32.2074i	−0.275268	+	1.91453i	0.114135	+	0.993465i	$0.463590\pi$
$283$	−4.63073	+	32.2074i	−0.275268	+	1.91453i	−0.389404	+	0.921067i	$0.627319\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.1102	+	7.78985i	−0.888835	+	0.458227i
$290$		0			0
$291$	27.6466	+	5.32844i	1.62067	+	0.312359i
$292$		0			0
$293$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$293$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	43.8877	+	34.5137i	2.52965	+	1.98933i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	5.76274	−	8.09264i	0.328897	−	0.461871i	−0.616706	−	0.787193i	$-0.711533\pi$
$307$	5.76274	−	8.09264i	0.328897	−	0.461871i	0.945603	+	0.325322i	$0.105473\pi$
$308$		0			0
$309$	0.696288	+	0.730246i	0.0396105	+	0.0415423i
$310$		0			0
$311$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$311$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$312$		0			0
$313$	19.4379	−	16.8430i	1.09869	−	0.952024i	0.0996196	−	0.995026i	$-0.468237\pi$
$313$	19.4379	−	16.8430i	1.09869	−	0.952024i	0.999075	+	0.0430013i	$0.0136920\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$317$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	26.8786	+	9.30279i	1.49096	+	0.516026i
$326$		0			0
$327$	−12.6331	−	27.6626i	−0.698611	−	1.52974i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−34.8929	−	8.46493i	−1.91789	−	0.465274i	−0.997182	−	0.0750153i	$-0.976099\pi$
$331$	−34.8929	−	8.46493i	−1.91789	−	0.465274i	−0.920705	−	0.390259i	$-0.872385\pi$
$332$		0			0
$333$	−5.06524	+	3.98335i	−0.277574	+	0.218286i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12.1518	+	15.4523i	0.661952	+	0.841740i	0.994864	−	0.101218i	$-0.0322738\pi$
$337$	12.1518	+	15.4523i	0.661952	+	0.841740i	−0.332913	+	0.942958i	$0.608031\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−23.5171	+	10.7399i	−1.26980	+	0.579900i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$347$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$348$		0			0
$349$	24.8239	−	7.28896i	1.32879	−	0.390169i	0.461136	−	0.887330i	$-0.347442\pi$
$349$	24.8239	−	7.28896i	1.32879	−	0.390169i	0.867659	+	0.497161i	$0.165624\pi$
$350$		0			0
$351$	26.2729	+	13.5446i	1.40234	+	0.722959i
$352$		0			0
$353$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$353$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$359$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$360$		0			0
$361$	30.9256	−	29.4875i	1.62766	−	1.55197i
$362$		0			0
$363$	15.5198	+	11.0516i	0.814576	+	0.580057i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	6.47592	+	33.6003i	0.338040	+	1.75392i	0.610751	+	0.791823i	$0.290868\pi$
$367$	6.47592	+	33.6003i	0.338040	+	1.75392i	−0.272711	+	0.962096i	$0.587920\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	25.0593	+	14.4680i	1.29752	+	0.749124i	0.979975	−	0.199119i	$-0.0638080\pi$
$373$	25.0593	+	14.4680i	1.29752	+	0.749124i	0.317546	+	0.948243i	$0.397141\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	6.60447	−	34.2673i	0.339249	−	1.76019i	−0.265824	−	0.964022i	$-0.585644\pi$
$379$	6.60447	−	34.2673i	0.339249	−	1.76019i	0.605073	−	0.796170i	$-0.293144\pi$
$380$		0			0
$381$	−2.48881	−	4.82762i	−0.127506	−	0.247326i
$382$		0			0
$383$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$383$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−37.2511	−	5.35591i	−1.89358	−	0.272256i
$388$		0			0
$389$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$389$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	32.7668	−	21.0579i	1.64452	−	1.05687i	0.708023	−	0.706189i	$-0.249587\pi$
$397$	32.7668	−	21.0579i	1.64452	−	1.05687i	0.936496	−	0.350679i	$-0.114049\pi$
$398$		0			0
$399$	39.6633	−	45.7739i	1.98565	−	2.29156i
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−25.5415			−1.27231
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	11.9679	−	29.8945i	0.591777	−	1.47819i	−0.265902	−	0.964000i	$-0.585670\pi$
$409$	11.9679	−	29.8945i	0.591777	−	1.47819i	0.857678	−	0.514186i	$-0.171906\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.44589	−	23.9667i	0.168746	−	1.17366i
$418$		0			0
$419$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$419$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$420$		0			0
$421$	−14.3493	+	5.74460i	−0.699342	+	0.279974i	−0.693963	−	0.720011i	$-0.744137\pi$
$421$	−14.3493	+	5.74460i	−0.699342	+	0.279974i	−0.00537983	+	0.999986i	$0.501712\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	20.2713	−	44.3879i	0.980996	−	2.14808i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−1.46854	+	15.3793i	−0.0705736	+	0.739080i	0.889053	+	0.457804i	$0.151364\pi$
$433$	−1.46854	+	15.3793i	−0.0705736	+	0.739080i	−0.959627	+	0.281276i	$0.909242\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	11.8484	+	20.5221i	0.565494	+	0.979465i	0.997004	+	0.0773561i	$0.0246478\pi$
$439$	11.8484	+	20.5221i	0.565494	+	0.979465i	−0.431509	+	0.902108i	$0.642019\pi$
$440$		0			0
$441$	22.2896	−	31.3013i	1.06141	−	1.49054i
$442$		0			0
$443$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$443$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$449$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−19.0727	+	36.9958i	−0.896112	+	1.73821i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.57665	+	5.02271i	0.167309	+	0.234952i	0.889715	−	0.456517i	$-0.150903\pi$
$457$	3.57665	+	5.02271i	0.167309	+	0.234952i	−0.722406	+	0.691469i	$0.756964\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$461$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$462$		0			0
$463$	−14.3572	−	0.683919i	−0.667237	−	0.0317844i	−0.288769	−	0.957399i	$-0.593246\pi$
$463$	−14.3572	−	0.683919i	−0.667237	−	0.0317844i	−0.378468	+	0.925614i	$0.623549\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$467$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$468$		0			0
$469$	36.4195	−	0.902802i	1.68169	−	0.0416875i
$470$		0			0
$471$	−24.1604	−	30.7225i	−1.11325	−	1.41562i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.86923	+	39.2399i	−0.0857660	+	1.80045i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$479$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$480$		0			0
$481$	9.95323	−	7.08766i	0.453828	−	0.323170i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	30.0712	−	31.5377i	1.36265	−	1.42911i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	30.0712	−	31.5377i	1.36265	−	1.42911i	0.796225	−	0.605000i	$-0.206827\pi$
$488$		0			0
$489$	−41.6583	+	14.4181i	−1.88386	+	0.652009i
$490$		0			0
$491$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$491$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	36.4247	−	21.0298i	1.63059	−	0.941423i	0.646682	−	0.762760i	$-0.276156\pi$
$499$	36.4247	−	21.0298i	1.63059	−	0.941423i	0.983910	−	0.178663i	$-0.0571771\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$503$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−29.0402	−	16.7664i	−1.28972	−	0.744621i
$508$		0			0
$509$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$509$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$510$		0			0
$511$	10.2791	+	4.69430i	0.454720	+	0.207664i
$512$		0			0
$513$	−7.72628	+	40.0878i	−0.341124	+	1.76992i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$521$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$522$		0			0
$523$	−45.5065	+	4.34534i	−1.98986	+	0.190009i	−0.991792	+	0.127861i	$0.959189\pi$
$523$	−45.5065	+	4.34534i	−1.98986	+	0.190009i	−0.998067	+	0.0621473i	$0.980205\pi$
$524$		0			0
$525$	1.83401	+	38.5006i	0.0800428	+	1.68031i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.3525	+	8.54824i	0.928368	+	0.371662i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−23.2854	−	36.2327i	−1.00112	−	1.55777i	−0.818438	−	0.574595i	$-0.805160\pi$
$541$	−23.2854	−	36.2327i	−1.00112	−	1.55777i	−0.182678	−	0.983173i	$-0.558477\pi$
$542$		0			0
$543$	17.2788	−	43.1603i	0.741504	−	1.85219i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−44.5388	+	2.12164i	−1.90434	+	0.0907150i	−0.966549	−	0.256481i	$-0.917437\pi$
$547$	−44.5388	+	2.12164i	−1.90434	+	0.0907150i	−0.937793	+	0.347196i	$0.887134\pi$
$548$		0			0
$549$	3.12659	+	32.7431i	0.133440	+	1.39744i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	17.6931	+	72.9318i	0.752385	+	3.10138i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$557$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$558$		0			0
$559$	70.0723	+	13.5053i	2.96374	+	0.571215i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$563$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−3.80760	+	39.8750i	−0.159904	+	1.67459i
$568$		0			0
$569$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$569$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$570$		0			0
$571$	46.9260	−	9.04424i	1.96379	−	0.378490i	0.980347	−	0.197280i	$-0.0632107\pi$
$571$	46.9260	−	9.04424i	1.96379	−	0.378490i	0.983444	−	0.181210i	$-0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	6.04672	+	6.34161i	0.251728	+	0.264005i	0.837458	−	0.546501i	$-0.184041\pi$
$577$	6.04672	+	6.34161i	0.251728	+	0.264005i	−0.585730	+	0.810506i	$0.699192\pi$
$578$		0			0
$579$	−7.59232	+	25.8571i	−0.315526	+	1.07458i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$587$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$588$		0			0
$589$	−9.93869	−	33.8481i	−0.409517	−	1.39469i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$593$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−39.1953	−	1.86710i	−1.60416	−	0.0764153i
$598$		0			0
$599$		0			0		−0.971812	−	0.235759i	$-0.924242\pi$
$599$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$600$		0			0
$601$	−31.6662	+	24.9025i	−1.29169	+	1.01580i	−0.293481	+	0.955965i	$0.594814\pi$
$601$	−31.6662	+	24.9025i	−1.29169	+	1.01580i	−0.998209	+	0.0598308i	$0.980944\pi$
$602$		0			0
$603$	−20.9554	+	12.8013i	−0.853369	+	0.521308i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2.85662	−	11.7751i	0.115947	−	0.477938i	−0.883999	−	0.467489i	$-0.845159\pi$
$607$	2.85662	−	11.7751i	0.115947	−	0.477938i	0.999945	−	0.0104491i	$-0.00332612\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8.57890	+	24.7871i	−0.346498	+	1.00114i	0.628284	+	0.777984i	$0.283758\pi$
$613$	−8.57890	+	24.7871i	−0.346498	+	1.00114i	−0.974782	+	0.223157i	$0.928364\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$617$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$618$		0			0
$619$	−35.1842	−	18.1387i	−1.41417	−	0.729056i	−0.429433	−	0.903099i	$-0.641287\pi$
$619$	−35.1842	−	18.1387i	−1.41417	−	0.729056i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−16.3715	−	18.8937i	−0.654861	−	0.755750i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	40.0222	+	28.4997i	1.59326	+	1.13455i	0.917720	+	0.397227i	$0.130027\pi$
$631$	40.0222	+	28.4997i	1.59326	+	1.13455i	0.675537	+	0.737326i	$0.263912\pi$
$632$		0			0
$633$	37.9992	−	21.9388i	1.51033	−	0.871990i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−45.0417	+	57.2752i	−1.78462	+	2.26933i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−24.8979	−	16.0009i	−0.981877	−	0.631014i	−0.0519076	−	0.998652i	$-0.516530\pi$
$643$	−24.8979	−	16.0009i	−0.981877	−	0.631014i	−0.929969	+	0.367638i	$0.880167\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$647$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−12.8641	−	32.1331i	−0.504185	−	1.25939i
$652$		0			0
$653$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$653$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−7.58245	+	0.724037i	−0.295820	+	0.0282474i
$658$		0			0
$659$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$659$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$660$		0			0
$661$	−44.6368	+	6.41780i	−1.73617	+	0.249623i	−0.936462	−	0.350768i	$-0.885921\pi$
$661$	−44.6368	+	6.41780i	−1.73617	+	0.249623i	−0.799706	+	0.600391i	$0.795011\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			44.2145i			1.70943i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	38.2430	+	33.1377i	1.47416	+	1.27736i	0.881742	+	0.471732i	$0.156371\pi$
$673$	38.2430	+	33.1377i	1.47416	+	1.27736i	0.592416	+	0.805632i	$0.298174\pi$
$674$		0			0
$675$	−14.0463	−	21.8564i	−0.540641	−	0.841254i
$676$		0			0
$677$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$677$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$678$		0			0
$679$	−10.2963	−	71.6122i	−0.395135	−	2.74822i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$683$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−9.30284	−	38.3468i	−0.354925	−	1.46302i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−39.0577	+	20.1356i	−1.48582	+	0.765996i	−0.994476	−	0.104967i	$-0.966526\pi$
$691$	−39.0577	+	20.1356i	−1.48582	+	0.765996i	−0.491349	+	0.870963i	$0.663496\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$701$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$702$		0			0
$703$	13.2657	+	10.4322i	0.500324	+	0.393459i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	12.9033	−	18.1202i	0.484595	−	0.680519i	−0.497809	−	0.867287i	$-0.665862\pi$
$709$	12.9033	−	18.1202i	0.484595	−	0.680519i	0.982404	+	0.186768i	$0.0598012\pi$
$710$		0			0
$711$	−34.9080	−	36.6105i	−1.30915	−	1.37300i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$719$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$720$		0			0
$721$	1.18806	−	2.30452i	0.0442458	−	0.0858248i
$722$		0			0
$723$	−8.08618	−	27.5390i	−0.300728	−	1.02419i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−7.37071	−	2.55103i	−0.273364	−	0.0946123i	0.186947	−	0.982370i	$-0.440141\pi$
$727$	−7.37071	−	2.55103i	−0.273364	−	0.0946123i	−0.460311	+	0.887758i	$0.652262\pi$
$728$		0			0
$729$	−11.2162	−	24.5601i	−0.415415	−	0.909632i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−50.9706	−	12.3653i	−1.88264	−	0.456724i	−0.882728	−	0.469885i	$-0.844295\pi$
$733$	−50.9706	−	12.3653i	−1.88264	−	0.456724i	−0.999913	+	0.0131610i	$0.995811\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	30.0719	+	38.2396i	1.10621	+	1.40667i	0.904109	+	0.427302i	$0.140536\pi$
$739$	30.0719	+	38.2396i	1.10621	+	1.40667i	0.202105	+	0.979364i	$0.435222\pi$
$740$		0			0
$741$	18.2509	−	75.2312i	0.670463	−	2.76369i
$742$		0			0
$743$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$743$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−43.1905	+	12.6819i	−1.57604	+	0.462768i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−43.1905	+	12.6819i	−1.57604	+	0.462768i	−0.627290	+	0.778785i	$0.715836\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	22.3615	−	7.73939i	0.812743	−	0.281293i	0.111088	−	0.993811i	$-0.464566\pi$
$757$	22.3615	−	7.73939i	0.812743	−	0.281293i	0.701654	+	0.712517i	$0.252445\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$761$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$762$		0			0
$763$	−56.5555	+	53.9255i	−2.04745	+	1.95224i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	5.77611	+	29.9693i	0.208292	+	1.08072i	0.924021	+	0.382343i	$0.124883\pi$
$769$	5.77611	+	29.9693i	0.208292	+	1.08072i	−0.715729	+	0.698378i	$0.753905\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$773$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$774$		0			0
$775$	19.4421	+	11.2249i	0.698380	+	0.403210i
$776$		0			0
$777$	13.9298	+	8.95212i	0.499728	+	0.321156i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	48.1050	−	11.6702i	1.71476	−	0.415996i	0.746388	−	0.665511i	$-0.231786\pi$
$787$	48.1050	−	11.6702i	1.71476	−	0.415996i	0.968371	+	0.249515i	$0.0802713\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.96768	−	62.2992i	−0.105385	−	2.21231i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$797$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$809$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$810$		0			0
$811$	−1.86653	+	4.66237i	−0.0655427	+	0.163718i	−0.957523	−	0.288356i	$-0.906891\pi$
$811$	−1.86653	+	4.66237i	−0.0655427	+	0.163718i	0.891981	+	0.452074i	$0.149316\pi$
$812$		0			0
$813$	7.62248	+	53.0155i	0.267332	+	1.85934i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	9.36896	+	98.1162i	0.327778	+	3.43265i
$818$		0			0
$819$	10.8095	−	75.1818i	0.377715	−	2.62706i
$820$		0			0
$821$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$821$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$822$		0			0
$823$	−24.0310	+	9.62056i	−0.837668	+	0.335352i	−0.750523	−	0.660844i	$-0.770198\pi$
$823$	−24.0310	+	9.62056i	−0.837668	+	0.335352i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$827$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$828$		0			0
$829$	−9.47962	+	20.7575i	−0.329241	+	0.720937i	−0.999781	−	0.0209367i	$-0.993335\pi$
$829$	−9.47962	+	20.7575i	−0.329241	+	0.720937i	0.670540	+	0.741873i	$0.266062\pi$
$830$		0			0
$831$	−24.6544	+	38.3630i	−0.855252	+	1.33080i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	18.3390	+	14.4220i	0.633889	+	0.498496i
$838$		0			0
$839$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$839$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	13.7930	−	46.9747i	0.473934	−	1.61407i
$848$		0			0
$849$	−42.5929	+	36.9070i	−1.46179	+	1.26664i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−20.2644	−	19.3220i	−0.693839	−	0.661574i	0.258712	−	0.965955i	$-0.416702\pi$
$853$	−20.2644	−	19.3220i	−0.693839	−	0.661574i	−0.952551	+	0.304381i	$0.901551\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$857$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$858$		0			0
$859$	27.0260	+	37.9528i	0.922117	+	1.29493i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	27.0260	+	37.9528i	0.922117	+	1.29493i	−0.0332309	+	0.999448i	$0.510580\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$863$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−28.6149	−	6.94189i	−0.971812	−	0.235759i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	40.8455	−	22.3556i	1.38400	−	0.757489i
$872$		0			0
$873$	30.1455	+	38.3331i	1.02027	+	1.29738i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.644251	−	13.5245i	0.0217548	−	0.456690i	−0.961539	−	0.274667i	$-0.911432\pi$
$877$	0.644251	−	13.5245i	0.0217548	−	0.456690i	0.983294	−	0.182023i	$-0.0582646\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$881$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$882$		0			0
$883$	26.4888	−	18.8626i	0.891421	−	0.634778i	−0.0398103	−	0.999207i	$-0.512675\pi$
$883$	26.4888	−	18.8626i	0.891421	−	0.634778i	0.931231	+	0.364429i	$0.118736\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$887$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$888$		0			0
$889$	−9.63117	+	10.1009i	−0.323019	+	0.338773i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	18.3017	+	94.9580i	0.609041	+	3.16000i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	9.26116	+	0.884334i	0.307512	+	0.0293638i	0.247671	−	0.968844i	$-0.420335\pi$
$907$	9.26116	+	0.884334i	0.307512	+	0.0293638i	0.0598406	+	0.998208i	$0.480941\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$911$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−20.1965	−	50.4485i	−0.666222	−	1.66414i	−0.744846	−	0.667237i	$-0.767477\pi$
$919$	−20.1965	−	50.4485i	−0.666222	−	1.66414i	0.0786243	−	0.996904i	$-0.474947\pi$
$920$		0			0
$921$	16.7225	−	4.05684i	0.551025	−	0.133677i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−10.6912	+	1.02089i	−0.351524	+	0.0335665i
$926$		0			0
$927$	0.0831559	+	1.74566i	0.00273120	+	0.0573349i
$928$		0			0
$929$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$929$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$930$		0			0
$931$	−93.4286	−	37.4032i	−3.06200	−	1.22584i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	58.0555i		−	1.89659i	−0.317391	−	0.948295i	$-0.602807\pi$
$937$		−	58.0555i		−	1.89659i	0.317391	−	0.948295i	$-0.397193\pi$
$938$		0			0
$939$	44.5484			1.45378
$940$		0			0
$941$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$941$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$947$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$948$		0			0
$949$	14.4269	−	0.687237i	0.468316	−	0.0223087i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$953$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	10.6444	+	2.05154i	0.343368	+	0.0661787i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	15.3918	−	26.6593i	0.494966	−	0.857306i	−0.505017	−	0.863109i	$-0.668514\pi$
$967$	15.3918	−	26.6593i	0.494966	−	0.857306i	0.999983	+	0.00580331i	$0.00184726\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$971$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$972$		0			0
$973$	−61.0943	+	11.7750i	−1.95859	+	0.377488i
$974$		0			0
$975$	24.6323	+	42.6645i	0.788866	+	1.36636i
$976$		0			0
$977$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$977$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	14.8397	−	50.5393i	0.473795	−	1.61360i
$982$		0			0
$983$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$983$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−3.97137	−	13.5252i	−0.126155	−	0.429644i	0.872058	−	0.489402i	$-0.162785\pi$
$991$	−3.97137	−	13.5252i	−0.126155	−	0.429644i	−0.998213	+	0.0597587i	$0.980967\pi$
$992$		0			0
$993$	−36.0733	−	50.6579i	−1.14475	−	1.60758i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−22.2964	−	48.8222i	−0.706133	−	1.54621i	−0.832373	−	0.554215i	$-0.813018\pi$
$997$	−22.2964	−	48.8222i	−0.706133	−	1.54621i	0.126241	−	0.992000i	$-0.459709\pi$
$998$		0			0
$999$	−11.1485	−	0.531069i	−0.352723	−	0.0168023i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.ba.a.281.1	✓	20
3.2	odd	2	CM	804.2.ba.a.281.1	✓	20
67.31	odd	66	inner	804.2.ba.a.701.1	yes	20
201.98	even	66	inner	804.2.ba.a.701.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.ba.a.281.1	✓	20	1.1	even	1	trivial
804.2.ba.a.281.1	✓	20	3.2	odd	2	CM
804.2.ba.a.701.1	yes	20	67.31	odd	66	inner
804.2.ba.a.701.1	yes	20	201.98	even	66	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.ba (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(1.30900 + 1.13425i) q^{3} +(1.65416 - 4.13190i) q^{7} +(0.426945 + 2.96946i) q^{9} +O(q^{10})\)	\(q+(1.30900 + 1.13425i) q^{3} +(1.65416 - 4.13190i) q^{7} +(0.426945 + 2.96946i) q^{9} +(-0.540736 - 5.66284i) q^{13} +(7.29407 - 2.92011i) q^{19} +(6.85191 - 3.53240i) q^{21} +(-2.07708 + 4.54816i) q^{25} +(-2.80925 + 4.37128i) q^{27} +(0.426797 - 4.46962i) q^{31} +(1.07398 + 1.86019i) q^{37} +(5.71527 - 8.02597i) q^{39} +(-3.53426 + 12.0366i) q^{43} +(-9.27019 - 8.83911i) q^{49} +(12.8601 + 4.45091i) q^{57} +(10.9516 + 0.521689i) q^{61} +(12.9758 + 3.14788i) q^{63} +(3.22957 + 7.52130i) q^{67} +(-0.120810 + 2.53611i) q^{73} +(-7.87764 + 3.59760i) q^{75} +(-13.7353 + 9.78084i) q^{79} +(-8.63544 + 2.53559i) q^{81} +(-24.2927 - 7.13299i) q^{91} +(5.62836 - 5.36663i) q^{93} +(14.0777 - 8.12776i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 20 * q + 6 * q^7 + 6 * q^9	\( 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) \) 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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